Natural catastrophes attract regularly the media attention and have become a source of public concern. From a financial viewpoint, they represent idiosyncratic risks, diversifiable at the world level. But for various reasons, reinsurance markets are unable to cope with this risk completely. Insurance-linked securities, such as catastrophe (cat) bonds, have been issued to complete the international risk transfer process, but their development is disappointing so far. This article argues that downside risk aversion and ambiguity aversion explain their limited success. Hybrid cat bonds, combining the transfer of cat risk with protection against a stock market crash, are proposed to complete the market. The article shows that replacing simple cat bonds with hybrid cat bonds would lead to an increase in market volume.